## Exponential Functions $f(x)=a^{x}$

### Exponential Function Rules

There are a number of rules that we can apply to any basic exponential function such as this.

Number one: $f(-1)=\frac{1}{a}$

Number two: $f(0)=1$

Number three: $f(1)=a$

Number four is that the x axis is a horizontal asymptote. $x \rightarrow \infty^{-}, f(x) \rightarrow 0$

Number five is that the domain includes all real numbers.

Number six is that the range of the function is all positive real numbers exclusive of zero

Number seven is that it is only a one to function, which means that every point on the x axis corresponds to one and only one point on the y axis

If we follow these rules the basic function is not hard at all to sketch.

## The Exponential Function

The main exponential function is euler’s number to the power of x ,which is known as the exponential function $f(x)=e^{x}$

To graph this we can just follow the same rules as the exponential functions.

## Transformations

### Translations $y=a^{x-h}+k$

• h represents the shift along the x-axis
• k the shift along the y-axis

So the rules above will face a shift along this axis $(-1,\frac{1}{a}) \rightarrow (-1+h,\frac{1}{a}+k)$ $(0,1) \rightarrow (h,1+k)$ $(1,a) \rightarrow (1+h,a+k)$

### Reflections

The graph can be reflected in the x-axis by placing a negative in front of the function to get: $f(x)=-e^{x}$

The graph can be reflected in the y-axis by placing a negative in front of x in the function to get: $f(x)=e^{-x}$

Here we can see the black function is a reflection of the blue function in the x axis and the red function is a reflection of the blue function in the x axis. It is also possible for a function to be reflected in both the x and y axis.

If there are translations applied to the function as well, you must first reflect the function before you translate them.

### Dialation

If the function has a dilation factor in front of it, say k. Then the image is multiplied by this factor. $(-1,\frac{1}{a}) \rightarrow (-1,\frac{k}{a})$ $(0,1) \rightarrow (0,k)$ $(1,a) \rightarrow (1,ka)$

If the function has a dilation factor of x divided by k. Then the image is dilated by a factor k from the y axis. $(-1,\frac{1}{a}) \rightarrow (-k,\frac{1}{a})$ $(0,1) \rightarrow (0,1)$ $(1,a) \rightarrow (k,ka)$

## Exponential functions Example

Lets now have a look at an example, such as the function: $f(x)=e^{1-x}+3$

This we can rewrite quickly as: $f(x)=e^{-(x-1)}+3$

Straight away we can see that there is a reflection in the y-axis, no dilation, but a shift of the graph of one to the right and three up.

Now we just need to apply this to our basic e function $(-1,\frac{1}{e}) \rightarrow (0,e+3)$ $(0,1) \rightarrow (1,4)$ $(1,e) \rightarrow (2,\frac{1}{e}+3)$

The asymptote will move up to x=3 